and 



X = 



- 17 



Here sgn (x) = - 1 according as x \ 0, respectively. With the substitution q = — tan 

 it follows that 



7 -£ f d ± 



n 2 J n <? 



In 



1 + j 



1 - 2 

 This leads to the simple result that 



sgn [(1 - ^) (/8 - t,)] = sgn [(1 - £) (|8 - »,)] 



?0 







10 <1 



10 >1 



and 

 and/or 



hi >/3 



that is, as one would expect, the pressure p is zero everywhere except directly under the rec- 

 tangle where it takes the value p Q . 



NUMERICAL RESULTS 



As stated before the integrals presented in Equation [2] can only be evaluated 

 numerically. A program for this purpose was written for the IBM 7090 and several examples 

 are given graphically in Figures 2 through 11. Details of the numerical analysis are outlined 

 in the Appendix. All of the calculations are for a beam-length ratio (/3) of 0.3. Table 1 

 shows the values of the remaining parameters pertinent to the various figures. 



In order to give some dimensional concept to the figures one can assume the rectangle 

 to be 100 feet long and 30 feet wide. With this configuration, Table 2 shows the dimensions 

 equivalent to the parameters of Table 1. 



For subcritical speed* (F < 1.0) there is both a transverse and divergent wave system 

 propagated behind the rectangle. As would be expected the oscillations of the centerline 

 (y/L = 0) pressure traces of Figures 2, 3, 6, and 9 are similar to those of the transverse 

 wave system. For supercritical speeds there are no transverse waves and, consequently, 

 one would not anticipate an oscillatory behavior of the centerline trace far downstream. 

 Figure 8 shows a comparison of centerline pressures for subcritical and supercritical speeds. 

 Off the centerline, however, the pressure trace can still exhibit an oscillatory pattern due 

 to the diverging wave system. Since the diverging waves are, in general, of shorter length 

 than the transverse waves, the extent to which they affect the bottom pressure depends very 



♦Critical speed 



fgh 



